Question

In order to test the water quality in a motel swimming pool, the tip of an ordinary syringe is dipped just below the surface of the water such that the barrel of the syringe makes an angle θ with the horizontal, as shown. The plunger is drawn backwards at a constant rate for a distance d in a time interval Δt. At the moment that the barrel of the syringe is filled to the length d, as indicated in the drawing, the plunger continues being drawn back at the same constant rate.

a. For this moment, enter an expression for the gauge pressure at the point where the water in the barrel meets the seal at the lower-end of the plunger. Neglect the diameter of the syringe. In addition to the variables from the problem statement, you may also include the density of water, Pw, and the magnitude of the acceleration due to gravity, g.

Answer 1

The gauge pressure at the point where the water in the barrel meets the seal at the lower-end of the plunger can be calculated using the equation P = Pwgh, where P is the gauge pressure, Pw is the density of water, g is the acceleration due to gravity, and h is the depth of the water.

Step-by-step explanation:

The gauge pressure (Pgauge) at a given depth in a fluid can be calculated using the equation:

Pgauge =Patm +ρ⋅g⋅h

where:

Patm is the atmospheric pressure,

ρ is the density of the fluid (water in this case),

g is the acceleration due to gravity,

h is the depth of the point below the surface.

In this case, the depth (h) is related to the geometry of the syringe. Let's define some variables:

L is the length of the barrel of the syringe filled with water,

d is the distance the plunger is drawn backward.

At the moment when the barrel is filled to the length d, the depth (h) is equal to L−d.

Therefore, the gauge pressure at the point where the water in the barrel meets the seal at the lower end of the plunger is:

Pgauge = Patm +ρ⋅g⋅(L−d)

Include the density of water (ρ) and the magnitude of the acceleration due to gravity (g) in your expression.

Answer 2

The gauge pressure in the syringe barrel when sampling water from a swimming pool can be expressed as P = hρg, assuming the syringe's diameter is negligible. The angle of the syringe with the horizontal and the length of water in the barrel do not directly affect the gauge pressure calculation.

Step-by-step explanation:

To determine the gauge pressure at the point where the water in the barrel meets the seal at the lower-end of the plunger in a motel swimming pool syringe test, we need to consider the following:

The gauge pressure at a certain depth in a fluid can be found using the formula P = hρg, where h is the depth, ρ (rho) is the density of the fluid (in this case, water), and g is the acceleration due to gravity.

The angle θ of the syringe with the horizontal does not directly affect the pressure calculation since the depth h below the water surface is what causes pressure, not the length of water filled in the syringe barrel. The depth h is not specified in the problem, but if we consider d to be the vertical distance from the surface to the tip of the syringe, we could express h in terms of d and θ. However, since only an expression needs to be provided and additional variables such as θ and d are not necessary for the basic pressure calculation given the problem's parameters, such expression would not be required.

As such, if we ignore the diameter of the syringe which suggests we consider it to be infinitesimally small, the simplest expression for the gauge pressure at the moment the barrel of the syringe is filled to the length d is simply P = hρg, with h being the depth at which the tip of the syringe is submerged.